nLab
crossed module

Idea

A crossed module (of groups) is:

• from the nPOV: a convenient way to encode a strict 2-group $G$ in terms of a morphism of two ordinary groups $\partial :G_2\to G_1$.

From other points of view it is:

• like the inclusion of a normal subgroup, but isn't an inclusion in general;
• like a module with a twisted ‘multiplication’;
• like the action of automorphisms on a group;
• a crossed complex concentrated in degrees $1$ and $2$;
• a nonabelian chain-complex;
• a Moore complex of certain simplicial groups.

Historically they were the first example of higher dimensional algebra to be studied.

Definition

Diagrammatic definition

A crossed module is

• a pair of groups $G_2,G_1$,

• morphisms of groups

  $$G_2\stackrel{\delta }{\to }{G}_1$$G_2 \stackrel{\delta }{\to}{G_1}

  and

  $$G_1\stackrel{\alpha }{\to }\mathrm{Aut}(G_2)$$G_1 \stackrel{\alpha}{\to} Aut(G_2)

  (which below we will conceive of as a map $\alpha :G_1\times G_2\to G_2$ analogous the adjoint action $\mathrm{Ad}:G\times G\to G$ of a group on itself)

• such that

  $$\begin{array}{ccccc}{G}_2\times G_2& & \stackrel{\delta \times \mathrm{Id}}{\to }& & G_1\times G_2\\ & {}_{\mathrm{Ad}}\searrow & & {\swarrow }_{\alpha }\\ & & G_2\end{array}$$\array{ G_2 \times G_2 &&\stackrel{\delta \times Id}{\to}&& G_1 \times G_2 \\ & {}_{Ad}\searrow && \swarrow_\alpha \\ && G_2 }

  and

  $$\begin{array}{ccc}{G}_1\times G_2& \stackrel{\alpha }{\to }& G_2\\ {\downarrow }^{\mathrm{Id}\times \delta }& & {\downarrow }^{\delta }\\ G_1\times G_1& \stackrel{\mathrm{Ad}}{\to }& G_1\end{array}$$\array{ G_1 \times G_2 &\stackrel{\alpha}{\to}& G_2 \\ \downarrow^{Id \times \delta} && \downarrow^{\delta} \\ G_1 \times G_1 &\stackrel{Ad}{\to}& G_1 }

  commute.

We may use the notation $(G_2,G_1,\delta )$, for this if the action is fairly obvious, including an explicit action, $(G_2,G_1,\delta ,\alpha )$, if there is a risk of confusion.

Definition in terms of equations

The two diagrams can be translated into equations, which may often be helpful.

• If we write the effect of acting with $g_1\in G_1$ on $g_2\in G_2$ as ${}^{g_1}{g}_2$, then the second diagram translates as the equation:

  $$\delta ({}^{g_1}{g}_2)=g_1\delta (g_2)g_1^{-1}.$$\delta({}^{g_1}g_2) = g_1\delta(g_2)g_1^{-1}.

  In other words, $\delta$ is equivariant for the action of $G_1$.

• The first diagram is slightly more subtle. The group $G_2$ can act on itself in two different ways, (i) by the usual conjugation action, ${}^{g_2}{g}_2^{\prime }=g_2{g}_2^{\prime }{g}_2^{-1}$ and (ii) by first mapping $g_2$ down to $G_1$ and then using the action of that group back on $G_2$. The first diagram says that the two actions coincide. Equationally this gives:

  $${}^{\delta (g_2)}{g}_2^{\prime }=g_2{g}_2^{\prime }{g}_2^{-1}.$${}^{\delta(g_2)}g^\prime_2 = g_2g^\prime_2g_2^{-1}.

This equation is known as the Peiffer rule in the literature.

Morphisms

For $G$ and $H$ two strict 2-groups coming from crossed modules $[G]$ and $H$, a morphism of strict 2-groups $f:G\to H$, and hence a morphism of crossed modules $[f]:[G]\to H$ is a 2-functor

$$Bf:BG\to BH$$\mathbf{B}f : \mathbf{B}G \to \mathbf{B}H

between the corresponding delooped 2-groupoids. Expressing this in terms of a diagram of the ordinary groups appearing in $[G]$ and $[H]$ yields a diagram called a butterfly. See there for more details.

Examples

• For $H$ any group, its automorphism crossed module is

  $$\mathrm{AUT}(H):=(G_2=H,G_1=\mathrm{Aut}(H),\delta =\mathrm{Ad},\alpha =\mathrm{Id}).$$AUT(H) := (G_2 = H, G_1 = Aut(H), \delta = Ad, \alpha = Id) \,.

  Under the equivalence of crossed modules with strict 2-groups this corresponds to the automorphism 2-group

  $${\mathrm{Aut}}_{\mathrm{Grpd}}(BH)$$Aut_{Grpd}(\mathbf{B}H)

  of automorphisms in the category Grpd of groupoids on the one-object delooping groupoid $BH$ of $H$.

• Almost the canonical example of a crossed module is given by a group $G$ and a normal subgroup $N$ of $G$. We take $G_2=N$, and $G_1=G$ with the action given by conjugation, whilst $\delta$ is the inclusion, $\mathrm{inc}:N\to G$. This is ‘almost canonical’, since if we replace the groups by simplicial groups $G_{.}$ and $N_{.}$, then $({\pi }_0(G_{.}),{\pi }_0(N_{.}),{\pi }_0(\mathrm{inc}))$ is a crossed module, and given any crossed module, $(C,P,\delta )$, there is a simplicial group $G_{.}$ and a normal subgroup $N_{.}$, such that the construction above gives the given crossed module up to isomorphism.

• Another standard example of a crossed module is $M{\to }^{0}P$ where $P$ is a group and $M$ is a $P$-module. Thus the category of modules over groups embeds in the category of crossed modules.

• If $\mu :M\to P$ is a crossed module with cokernel $G$, and $M$ is abelian, then the operation of $P$ on $M$ factors through $G$. In fact such crossed modules in which both $M$ and $P$ are abelian should not be sneezed at! A good example is $\mu :C_2\times C_2\to C_4$ where $C_n$ denotes the cyclic group of order $n$, $\mu$ is injective on each factor, and $C_4$ acts on the product by the twist. This crossed module has a classifying space $X$ with fundamental and second homotopy groups $C_2$ and non trivial $k$-invariant in $H^3(C_2,C_2)$, so $X$ is not a product of Eilenberg-MacLane spaces. However the crossed module is an algebraic model and so one one can do algebraic constructions with it. It gives in some ways a better feel for the space than the $k$-invariant. The higher homotopy van Kampen theorem implies that the above $X$ gives the 2-type of the mapping cone of the map of classifying spaces ${\mathrm{BC}}_2\to {\mathrm{BC}}_4$.

• Suppose $F\stackrel{i}{\to }E\stackrel{p}{\to }B$ is a fibration sequence

  of pointed spaces, thus $p$ is a fibration in the topological sense (lifting of paths and homotopies of paths will suffice), $F=p^{-1}(b_0)$, where $b_0$ is the basepoint of $B$. The fibre $F$ is pointed at $f_0$, say, and $f_0$ is taken as the basepoint of $E$ as well.

  There is an induced map on homotopy groups

  $${\pi }_1(F)\stackrel{{\pi }_1(i)}{⟶}{\pi }_1(E)$$\pi_1(F) \stackrel{\pi_1(i)}{\longrightarrow} \pi_1(E)

  and if $a$ is a loop in $E$ based at $f_0$, and $b$ a loop in $F$ based at $f_0$, then the composite path corresponding to $ab{a}^{-1}$ is homotopic to one wholly within $F$. To see this, note that $p(ab{a}^{-1})$ is null homotopic?. Pick a homotopy in $B$ between it and the constant map, then lift that homotopy back up to $E$ to one starting at $ab{a}^{-1}$. This homotopy is the required one and its other end gives a well defined element ${}^{a}b\in {\pi }_1(F)$ (abusing notation by confusing paths and their homotopy classes). With this action $({\pi }_1(F),\pi (E),{\pi }_1(i))$ is a crossed module. This will not be proved here, but is not that difficult. (Of course, secretly, this example is ‘really’ the same as the previous one since a fibration of simplicial groups is just morphism that is an epimorphism in each degree, and the fibre is thus just a normal simplicial subgroup. What is fun is that this generalises to ‘higher dimensions’.)

• A particular case of this last example can be obtained from the inclusion of a subspace $A\to X$ into a pointed space $(X,x_0)$, (where we assume $x_0\in A$). We can replace this inclusion by a homotopic fibration, $\overline{A}\to X$ in ‘the standard way’, and then find that the fundamental group of its fibre is ${\pi }_2(X,A,x_0)$.

A deep theorem of J.H.C. Whitehead is that the crossed module

$$\delta :{\pi }_2(A\cup \{e_{\lambda }^2{\}}_{\lambda \in \Lambda },A,x)\to {\pi }_1(A,x)$$\delta: \pi_2(A \cup \{e^2_\lambda\}_{\lambda \in \Lambda},A,x) \to \pi_1(A,x)

is the free crossed module on the characteristic maps of the $2$-cells. One utility of this is that it enables the expression of nonabelian chains and boundaries ideas in dimensions $1$ and $2$: thus for the standard picture of a Klein Bottle formed by identifications from a square $\sigma$ the formula

$$\delta \sigma =a+b-a+b$$\delta \sigma = a+b-a +b

makes sense with $\sigma$ a generator of a free crossed module; in the usual abelian chain theory we can write only $\partial \sigma =2b$, thus losing information.

Whitehad’s proof of this theorem used knot theory and transversality. The theorem is also a consequence of the $2$-dimensional Seifert-van Kampen Theorem, proved by Brown and Higgins, which states that the functor

${\Pi }_2$: (pairs of pointed spaces) $\to$ (crossed modules)

preserves certain colimits (see reference below).

This last example was one of the first investigated by Whitehead and his proof appears also in a little book by Hilton.

Related concepts

• group

• 2-group, crossed module, differential crossed module

• 3-group, 2-crossed module / crossed square, differential 2-crossed module

• n-group

• ∞-group, simplicial group, crossed complex, hypercrossed complex

References

• R. Brown, “Groupoids and crossed objects in algebraic topology”, Homology, Homotopy and Applications, 1 (1999) 1-78.

• R, Brown and P.J. Higgins, “On the connections between the second relative homotopy groups of some related spaces”, Proc. London Math. Soc. (3) 36 (1978) 193-212.